Stationary states in single-well potentials under symmetric Levy noises

TL;DR

This paper investigates the conditions under which stationary states exist in systems with single-well potentials influenced by symmetric Levy noises, revealing a critical relation between potential steepness and noise stability.

Contribution

It analytically derives the necessary condition for stationary states under symmetric Levy noise and characterizes their heavy-tailed probability densities.

Findings

Stationary states exist if the potential exponent c > 2 - alpha.

Stationary probability densities decay as x^{-(c+alpha-1)} at infinity.

Monte Carlo simulations confirm the analytical predictions.

Abstract

We discuss the existence of stationary states for subharmonic potentials $V(x) \propto |x|^c$, $c<2$, under action of symmetric $\alpha$-stable noises. We show analytically that the necessary condition for the existence of the steady state is $c>2-\alpha$. These states are characterized by heavy-tailed probability density functions which decay as $P(x) \propto x^{-(c+\alpha -1)}$ for $|x| \to \infty$, i.e. stationary states posses a heavier tail than the corresponding $\alpha$-stable law. Monte Carlo simulations confirm the existence of such stationary states and the form of the tails of corresponding probability densities.